Determining the shear viscosity of model liquids from molecular dynamics simulations Hess, Berk

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1 University of Groningen Determining the shear viscosity of model liquids from molecular dynamics simulations Hess, Berk Published in: Journal of Chemical Physics DOI: 1.163/ IMPORA OE: You are advised to consult the publisher's version (publisher's PDF) if you wish to cite from it. Please check the document version below. Document Version Publisher's PDF, also known as Version of record Publication date: 22 Link to publication in University of Groningen/UMCG research database Citation for published version (APA): Hess, B. (22). Determining the shear viscosity of model liquids from molecular dynamics simulations. Journal of Chemical Physics, 116(1), Copyright Other than for strictly personal use, it is not permitted to download or to forward/distribute the text or part of it without the consent of the author(s) and/or copyright holder(s), unless the work is under an open content license (like Creative Commons). ake-down policy If you believe that this document breaches copyright please contact us providing details, and we will remove access to the work immediately and investigate your claim. Downloaded from the University of Groningen/UMCG research database (Pure): For technical reasons the number of authors shown on this cover page is limited to 1 maximum. Download date:

2 Determining the shear viscosity of model liquids from molecular dynamics simulations Berk Hess Citation: J. Chem. Phys. 116, 29 (22); doi: 1.163/ View online: View able of Contents: Published by the American Institute of Physics Articles you may be interested in Comparison of simple potential functions for simulating liquid water he Journal of Chemical Physics 79, 926 (1983); 1.163/ Viscosity calculations of n-alkanes by equilibrium molecular dynamics he Journal of Chemical Physics 16, 9327 (1997); 1.163/ he shear viscosity of rigid water models he Journal of Chemical Physics 132, 9611 (21); 1.163/ Molecular dynamics with coupling to an external bath he Journal of Chemical Physics 81, 3684 (1984); 1.163/ Canonical sampling through velocity rescaling he Journal of Chemical Physics 126, 1411 (27); 1.163/ Comparison of constant pressure and constant volume nonequilibrium simulations of sheared model decane he Journal of Chemical Physics 1, 541 (1994); 1.163/

3 JOURAL OF CHEMICAL PHYSICS VOLUME 116, UMBER 1 1 JAUARY 22 Determining the shear viscosity of model liquids from molecular dynamics simulations Berk Hess a) Department of Biophysical Chemistry, University of Groningen, ijenborgh 4, 9747 AG Groningen, he etherlands Received 8 June 21; accepted 2 October 21 Several methods are available for calculating shear viscosities of liquids from molecular dynamics simulations. here are equilibrium methods based on pressure or momentum fluctuations and several nonequilibrium methods. For the nonequilibrium method using a periodic shear flow, all relevant quantities, including the accuracy, can be estimated before performing the simulation. We compared the applicability, accuracy and efficiency of this method with two equilibrium methods and another nonequilibrium method, using simulations of a Lennard-Jones fluid and the SPC and SPC/E extended simple point charge water models. 22 American Institute of Physics. DOI: 1.163/ I. IRODUCIO he shear viscosity is a property of liquids which can be determined easily by experiment. It is useful for parametrizing the force field, because it is a kinetic property, while most other properties which are used for parametrization are of a thermodynamic nature. It is not only important for pure liquids, as it also influences the rates of rotational diffusion and conformational change of molecules solvated in the liquid. Several methods for determining the shear viscosity are described in the literature. From an equilibrium simulation the viscosity can be obtained from pressure or momentum fluctuations. 1 here are two nonequilibrium methods that make use of steady-state shear. One can use a periodic shear flow, 2 or use sliding boundary conditions, for instance the commonly used SLLOD 3 algorithm. 4 All the methods mentioned above, except for the momentum fluctuation approach, are described in Allen and ildesley. 5 A recent method uses momentum pulse relaxation. 6 he authors claim that, unlike the periodic shear flow method, their method is not wavelength or box-size dependent since acoustic modes cannot travel across the box boundaries. However, acoustic modes are not the origin of the wavelength dependence. Steady shear does not induce density fluctuations in a liquid and thus also no acoustic waves. he momentum pulse relaxation method suffers from the same wavelength dependence as the periodic shear flow method, since in both methods the size of the perturbation is of the order of the simulation box. We will describe the two equilibrium methods, the periodic shear flow method and the SLLOD algorithm in more detail. We will show applications of these four methods to a Lennard-Jones fluid and two water models. a Electronic mail: hess@chem.rug.nl II. PRESSURE FLUCUAIOS he shear viscosity of a liquid is related to the fluctuations of the off-diagonal elements of the pressure or stress tensor. he viscosity can be calculated from an equilibrium simulation by integrating the Green Kubo formula: 7 V k B Pxz t P xz t t dt. 1 t his can be reformulated as an Einstein relation: 1 V d t lim t 2 k B dt t Pxz t tdt 2. 2 t Both methods converge very slowly, because the pressure in a volume of the size of a simulation box fluctuates heavily. he asymptotic behavior of the pressure autocorrelation is t 3/2, which makes an accurate estimation of the viscosity difficult. he Einstein relation is more convenient to use, since the inaccuracies in the long time correlations can be ignored by only considering integrals over shorter times. III. MOMEUM FLUCUAIOS he viscosity can also be determined from transversecurrent correlation functions for plane waves. he behavior of the velocity u(r,t) of a liquid is described by the avier Stokes equation: u t u u ap 2 u, 3 where a is the external force per unit of mass and volume. Consider an incompressible liquid with an initial velocity field consisting of a plane wave in one direction: cosk z ur,u /22/116(1)/29/9/$ American Institute of Physics

4 21 J. Chem. Phys., Vol. 116, o. 1, 1 January 22 Berk Hess he y and z-component of u will remain zero; for the x-component the avier Stokes equation reduces to u x z,t 2 u x z,t t z 2. 5 he solution is u x z,tu e t/ r cosk z, r k 2. 7 From the solution it can be seen that plane waves decay exponentially with a time constant which is inversely proportional to the viscosity and proportional to the wavelength squared. According to the fluctuation dissipation theorem the response of the system to internal fluctuations is the same as to a small external perturbation, such as initial condition 4. his means the viscosity can be determined from the autocorrelation of the amplitude of plane waves in the simulation box, which is also called the transverse-current correlation function. However, at short times the correlation function is not a pure exponential. o account for this, a phenomenological correction can be applied by incorporating a relaxation-time. 8,9 his changes Eq. 5 to 6 u x z,t t 2 u x z,s ts t z 2 ds. 8 he most simple model for the memory kernel is t 1 e t/ m. 9 m he solution of this equation with initial condition 4 is where u x z,tu e cosh 1 sinh cosk z, t 2 m and m k2. 12 ote that although can become imaginary, the solution will always be real. For large k the solution converges to the solution of the avier Stokes equation 6. he viscosity can be calculated by fitting the transverse-current correlation functions for different k-vectors to formula 1. Unfortunately the shear viscosity which is obtained using this method is dependent on kk. o obtain the macroscopic viscosity, one needs to extrapolate to k. Palmer argues that since the viscosity in one dimension should be a symmetric function of k, one can approximate the viscosity in third order by k1ak 2 Ok IV. HE PERIODIC PERURBAIO MEHOD Instead of measuring intrinsic fluctuations of the system, which have a fixed magnitude, we can apply an external force. he magnitude of this force can be chosen such that the effects are much easier to measure than the internal fluctuations. he external force will cause a velocity field u in the liquid according to the avier Stokes equation 3. We can choose the external force a such that a y and a z are zero and a x is a function of z only. his makes u y and u z equal to zero. Since there is no pressure gradient in the x-direction, the equation for u x becomes u xz a t x z 2 u x z z he steady-state solution is given by a x z 2 u x z z Since we will be using a periodic system in the simulations, the acceleration and velocity profile should also be periodic. o obtain a smooth velocity profile with small local shear rates, the acceleration profile should be smooth as well. A simple function which satisfies both conditions is a cosine: a x za cosk z, 16 k 2, 17 l z where l z is the height of the box. When we take u z (x) at t, the generated velocity profile is u x zv1e t/ rcosk z, 18 VA k 2, 19 where r is given by 7. he avier Stokes equation is not valid for microscopic length-scales, therefore the wavelength should be at least an order of magnitude larger than the size of a molecule. In an MD-simulation an acceleration can be added to each particle each MD step, according to Eq. 16. he average V can be measured and the viscosity can be calculated using the following formula: A V k 2. 2 We define the instantaneous V(t) in the simulation as follows: Vt2 m i v i,x tcosk r i,z i1 t i1 m i, 21 where v i,x is the x-component of the velocity, r i,z is the z-coordinate and m i is the mass of atom i. he average for V can be measured after the amplitude of the velocity profile has been fully developed, which is around t5 r. o obtain the correct value for the viscosity the generated velocity profile should not be coupled to the heat bath, also the velocity profile should be excluded from the kinetic energy. One

5 J. Chem. Phys., Vol. 116, o. 1, 1 January 22 Shear viscosity from molecular dynamics simulations 211 would like V to be as large as possible to get good statistics. However, the shear rate should not be so high that the system moves too far from equilibrium. he maximum shear rate occurs where the cosine is zero, the rate is: s max max v xza z z k kg m 1 s 1 k1 nm kg m 3 A.1 nm ps 2 C v 41 3 Jkg 1 K 1 V.1 nm ps 1 27 he system will be able to relax when the inverse shear rate is longer than the relaxation time. For most liquids this will be the rotation correlation time, which is around 1 picoseconds. When the shear rate is too high the apparent viscosity will be too low. he accuracy of the viscosity estimate is determined by the momentum fluctuations described in the previous section. he amplitude of the fluctuation on V(t) is VtVt 2 1/2 2k B V. 23 he 2 arises from the integral over (2 cos) 2 in 21; V is the total mass of the system. For large wavelengths the fluctuations have exponential correlation 1, with correlation time r 7. From this the error in the average can be calculated. he relative standard error in the amplitude of the velocity profile is V V 2 r t a 2k B s max V k 2 s max k B t a V, 24 where t a is the time over which the average of V(t) is determined. he error in the viscosity is V V 2 s max k B t a V. 25 he accuracy increases with the square root of time and the number of particles. When the CPU-time increases linearly with the number of particles, the CPU-time required for a given accuracy is independent of the system size. hus a large system is optimal, since it produces a longer wavelength. he system should not be chosen too large, because the equilibration and shear development time increase as well. he heat generated by the viscous friction is removed by coupling to a heat bath. Because this coupling is not instantaneous the real temperature of the liquid will be slightly lower than the observed temperature. Berendsen derived this temperature shift for coupling with the Berendsen thermostat. 1,11 We rewrote the shift in terms of the shear rate: s c 2 s 2 C max, 26 v where c is the coupling time for the Berendsen thermostat and C v is the heat capacity. o get an idea of the order of magnitude of the quantities involved, we can look at a small simulation system for water at 2 C: l z 2 nm c.1 ps s max 1/1 ps 1. r 1ps s.125 K hus for the requested shear rate we get a measurable value for V and a temperature shift of.125 K, which is negligible. V. HE SLLOD ALGORIHM One can also obtain the viscosity by imposing a Couette flow. he equation of motion needs to be modified to achieve this. In the SLLOD algorithm this is done in a non- Hamiltonian way. When shearing in the x-direction with the gradient in the z-direction, the SLLOD equations of motion are dr i dt v is r i,z1, dv i dt 1 f m i s v i,z1 i, where i is the particle index, a second index indicates the vector component and s is the shear rate. As the system is periodic, the periodic boundary conditions also need to be modified. his can be accomplished by using a continuously deforming triclinic unit-cell, which is implemented in the GROMACS 3. package, 12 that we modified to use the SLLOD method. he viscosity is calculated from the stress tensor P: 1 s P xz. 3 here are two problems with this approach, besides the equations of motion not being Hamiltonian. he first problem is that conceptually the viscosity does not correspond to the viscosity obtained from an experiment. In the SLLOD algorithm not a force profile, but the linear velocity profile is imposed up to the atomic level, while in the experiment external forces induce a Couette flow on the macroscopic level. On the atomic level the velocity profile is not necessarily linear. In simple liquids these viscosities will be the same, but in complex liquids they can be different. A second problem is that the equations of motion produce an overall rotation in the system.

6 212 J. Chem. Phys., Vol. 116, o. 1, 1 January 22 Berk Hess FIG. 1. Viscosity estimate for a Lennard-Jones fluid using Eq. 2. he derivative is plotted as a function of time for the whole simulation solid line and the two halves dashed lines. VI. VISCOSIY OF A LEARD-JOES FLUID As a simple model system we chose a fluid of Lennard- Jones particles. he system was made dimensionless by expressing units in, and the mass. All simulations have been performed at number density n*.452 and temperature *2, starting from an equilibrated conformation. he time step was.1, the neighbor list was updated every 1 steps, with a cut-off of 5. he temperature was coupled using a Berendsen thermostat with a coupling time of 2. For the pressure fluctuations method we performed a simulation in double precision with a duration of 5 and in a periodic, cubic box containing 1 particles. he right hand side of Eq. 2 averaged over the three off-diagonal elements, without taking the limit, is shown in Fig. 1. Since the curves are reasonably constant from t*5 to 8, we took the average over this period. An error estimate can be made from the three values for the off-diagonal elements. For the whole simulation the dimensionless viscosity * is For the transverse-current method we simulated two systems for a duration of 5: the system of 1 particles described above and 8 particles in a cubic box. he coordinates and velocities were written at intervals of.2. he k-vectors used for the analysis are 1,,, 1,1,, (1, 1,), (1,1,1), 2,, each in all 3 permutations and 1,1,1, for notational convenience we left out the inverse box length for each k-vector component. For each k-vector there are 2 perpendicular velocity directions and 2 amplitudes, one of sin(k r) and one of cos(k r). For each k-vector length there are only 3 independent directions. Since our simulation boxes are cubic, we can average over all permutations of the k-vectors given above, as well as over the k-vectors which differ only in signs. he transversecurrent autocorrelation functions are close to exponential, except for the initial decay not shown. A higher density would produce a negative minimum in the autocorrelation. he autocorrelation functions can be fitted very well with expression 1, where we used exponentially decaying weights with time constant 1. he resulting k-dependent viscosities of the 1 particle system can be fitted perfectly FIG. 2. Viscosity estimate for a Lennard-Jones fluid using transverse current autocorrelation functions. he circles are for the 8 particle system; the diamonds for the 1 particle system. he filled symbols are averages over all k-vectors open symbols with equal length. he solid lines are the fits to expression 13 for the separate systems; the dashed line is the fit for both systems. with the second order approximation 13, this gives *.433 Fig. 2. For the 8 particle system the fit is worse; the value is *.465. Fitting one curve to both systems results in *.448. It is hard to tell which of these values is the best one. *(k) will be closer to the real value of the viscosity for small k, but the error increases, since the correlation time is proportional to 1/k 2. For the periodic perturbation method we simulated systems of 3 different sizes, to study the wavelength dependence: 1 particles in a cubic box as described above and 2 and 4 of these boxes stacked in the z-direction. he simulation protocol was the same as for the other methods, except for the coupling time of the Berendsen thermostat, which was set to 5, to more rapidly remove the energy introduced in the system by the external force. For each system size we used 3 different amplitudes for the acceleration profile. All simulations have length 5. We started the analysis at time 4, which allows for enough time to develop a steady shear. Error estimates were calculated assuming a double exponential auto-correlation using formula A16, which is derived in the Appendix. one of the simulations show a significant long correlation time. he short correlation time is close to r Eq. 7. All results are shown in able I. he medium ABLE I. Viscosity measurements of a Lennard-Jones fluid of particles using the periodic perturbation method with 3 different system sizes. s max * and * are averages over time 4 5. wo error estimates are given for the viscosity, one based on block averaging see the Appendix and between parentheses expression 25. k* A* s max * * (.13) (.7) (.3) (.1) (.5) (.2) (.7) (.4) (.2)

7 J. Chem. Phys., Vol. 116, o. 1, 1 January 22 Shear viscosity from molecular dynamics simulations 213 and large systems give compatible values for the viscosity at all shear rates. he small system gives lower viscosities. his indicates the viscosity converges to the infinite wavelength value somewhere between k*.48 and.24. he shear-rate dependence is negligible, since the highest shear rate.17 is still 15 times smaller than the inverse of the exponential decay time of the autocorrelation 1/.4. he temperature shifts at s max.8 and.17 are.5% and 2%, respectively. he most accurate estimate of the viscosity can be obtained from the large system, averaging the values for the 3 shear rates gives * For the SLLOD method we used the 1 particle system with a shear rate of.5. he viscosity, averaged over time 4 5 is he temperature shift is.4%. o compare the accuracy at equal CPU-time, the runtime of the periodic perturbation simulations of 4 particles should be decreased by a factor of 4. aking into account the shear development time, the accuracy decreases by a factor 2.3, which makes the periodic perturbation method more accurate than both equilibrium methods at all 3 shear rates. At an equal average shear rate (s max 2s) the nonequilibrium methods are equally accurate. VII. VISCOSIY OF HE SPC AD SPCÕE WAER MODELS o estimate the viscosity of the SPC simple point charge water model 13 from the pressure fluctuations, we have performed MD-simulations in double precision of 1728 molecules in a cubic box of length 3.75 nm. All simulations were done at a constant density of 98 kg m 3. he temperature was coupled to 3 K, using a Berendsen thermostat 1 with a coupling time of.1 ps. he time step was 2 fs and the neighbor list, which is built with a cut-off of.9 nm, was updated every 5 steps. he cut-off for the Lennard-Jones interactions was.9 nm. he water geometry was maintained with the SELE algorithm. 14 We performed 5 simulations of 1 ns with different electrostatics treatments, starting from an equilibrated conformation. he first simulation used a Coulomb cut-off of.9 nm. he second simulation used a Coulomb cut-off of 1.4 nm, all forces between.9 and 1.4 nm were updated every 5 steps. he protocols for the third and fourth simulation are identical to the protocol of the first and second, respectively, except that a reaction field 15,16 with a dielectric constant of 8 was used. he fifth simulation used Particle Mesh Ewald PME 17 for the electrostatics, the cut-off for the particle particle interactions was.9 nm. Because the simulations were performed at a constant volume, the pressure ranges from 3 bar for the.9 nm cut-off simulation to 5 bar for the 1.4 nm cut-off reaction-field simulation. he pressure fluctuations are about 27 bar for all 5 simulations. Running at a constant pressure of 1 bar would change the density by 1 or 2%, which would have little effect on the pressure fluctuations. o obtain the viscosity, we calculated the expression of Eq. 2, without taking the limit. he results for the 5 simulations are shown in Fig. 3. he differences between the simulations are large. his reflects the dependence of the pressure fluctuations on the treatment of the electrostatics. he long-range Coulomb interactions FIG. 3. Viscosity estimate for SPC water using Eq. 2. he derivative is plotted as a function of time for 5 simulations which differ only in electrostatics treatment. have little effect on the potential, but the forces contribute significantly to the virial. Artifacts of the electrostatics cutoff can be seen in the dipole dipole correlation plot Fig. 4. At distances slightly below the cut-off the dipoles are anticorrelated, at distances slightly above the cut-off the dipoles are correlated. In addition to this effect, the simulation with a cut-off of 1.4 nm shows extra correlation at short distances. A reaction-field decreases the long-range correlations significantly. he reaction-field simulation with a cut-off of.9 nm has a lower dipole correlation in the first coordination sphere.28 nm. With PME there is no correlation after.75 nm. he too large correlation of dipoles at long distances makes an accurate estimation of the viscosity from pressure fluctuations impossible. A problem with all the simulations is that the derivative is not constant. Smith and Van Gunsteren applied the same method for a smaller system of 512 SPC water molecules using a cut-off of.9 nm. 18 hey obtained viscosities of (kgm 1 s 1 ) and with a reaction-field. o check the convergence, we performed a simulation of 2 ns with PME, with a time step of 4 fs. he viscosity results are shown in Fig. 5. Still the pressure fluctuations look far from converged. We chose to average over time 1 to FIG. 4. he average inner product between the water dipole directions as a function of the distance between the dipoles.

8 214 J. Chem. Phys., Vol. 116, o. 1, 1 January 22 Berk Hess FIG. 5. Viscosity estimate for SPC water using Eq. 2. he derivative is plotted as a function of time for a simulation of 2 ns using PME. he dotted lines are the three off-diagonal elements of the pressure tensor; the solid line is the average. 2 ps, because the three off-diagonal elements are relatively close over this period. he resulting value for the viscosity is (kgm 1 s 1 ). ote that even within these first 2 picoseconds the off-diagonal elements differ up to 1%. o estimate the viscosity from the transverse-current correlation functions, we simulated the same system as above for 2 ns with PME. We used a coupling time of 2.5 ps instead of.1 ps for the Berendsen thermostat to minimize the influence of the thermostat on the correlation functions. he coordinates and velocities were written each.1 ps. o check the dependence of the viscosity on the size of the simulation box, we also simulated a system of 512 SPC water molecules at the same density with the same simulation protocol, coordinates and velocities were written every.5 ps. We calculated transverse current correlation functions using the center of mass of the water molecules. he transverse currents were calculated for the same k-vectors as for the Lennard-Jones fluid. Figure 6 shows the transverse-current autocorrelation functions as well as the fits for the 4 different k-vector lengths for the 1728 SPC system. Although the fit is better than a pure exponential fit, the first part of the curves is FIG. 6. ransverse current autocorrelation functions for the 1728 SPC molecule system for 4 different k-vector lengths. he solid lines are the correlation functions; the dashed lines the fit to expression 1. FIG. 7. Viscosity estimate for SPC water using transverse current autocorrelation functions. he circles are for the 1728 molecule system; the diamonds for the 512 molecule system. he filled symbols are averages over all k-vectors open symbols with equal length. he line is the fit of the filled symbols to expression 13. steeper than the fit and the negative part of the curves is also not fitted well. he values of the viscosity obtained from both simulations Fig. 7 are consistent. Although a straight line fits the data much better, we fitted with the second order function 13, because the viscosity should be an even function of k. he resulting viscosity is (kgm 1 s 1 ). We also measured the viscosity for several simulations of SPC water with the periodic perturbation method, using 3 different system sizes and several electrostatics treatments. he simulation protocol was identical to the one used for the equilibrium simulations described previously, except that each atom was subject to an additional acceleration according to Eq. 16. he system sizes, the electrostatics treatment and the results are shown in able II. he accuracy of all simulations is in agreement with expression 25. he results for the smallest system show that the shear rate should not be higher than.1 ps 1 ; above this value the apparent viscosity starts to decrease and the temperature starts to increase. he simulations for the medium system size show the dependence of the viscosity on the electrostatics treatment and the temperature. he viscosity for the 2 cut-off simulations is 1% higher than PME simulation at 33 K and the reactionfield simulation with cut-off.9. he reaction-field simulation with twin-range cut-off is in between. he relative difference in the viscosity between the PME simulation at 3 and 33 K is two times as small as in experiment. For the large system the two cut-off simulations show that there is no dependence on the shear-rate at.26 ps 1. he PME simulation produces a viscosity of (kgm 1 s 1 ). he same value is obtained when the viscosities of the PME simulations at the 3 different wavelengths are fitted to expression 13. he cut-off and PME simulations show the same k-dependence. For the SLLOD method we simulated the 1728 SPC system mentioned above. he shear rate was.25 ps 1. he viscosity was obtained by averaging from 5 to 2 ps. For a cut-off of.9 nm the viscosity is (kgm 1 s 1 ), for PME electrostatics it is (kgm 1 s 1 ).

9 J. Chem. Phys., Vol. 116, o. 1, 1 January 22 Shear viscosity from molecular dynamics simulations 215 ABLE II. Viscosity measurements using the periodic perturbation method with 3 different system sizes and experiment Ref. 2. he first column shows the x/yz dimensions of the simulation box, the number of molecules and the period used for analysis. c-o indicates cut-off treatment of the electrostatics, R c is the cut-off distance. P pressure, m the measured temperature, s max and are averages over the period indicated in the first column. wo error estimates are given for the viscosity, one based on block averaging and between brackets expression 25. ote that m is a few degrees higher than the coupling temperature of 3 K for simulations without PME. Elec. R c nm A (nm ps 2 ) P bar m K s max (ps 1 ) (1 3 kg m 1 s 1 ) c-o (.26) nm c-o (.13) 648 SPC c-o (.6) 2 2 ps c-o (.2) PME (.12) c-o (.4) nm c-o (.4) 3456 SPC RF (.3) 2 1 ps RF (.4) PME (.3) PME (.3) nm c-o (.11) 6912 SPC c-o (.5) 5 2 ps PME (.6) nm (.9) 6912 SPC/E PME 5 2 ps experiment We performed one periodic perturbation simulation using a SPC/E extended simple point charge water model. 19 he simulation protocol is identical to the large SPC system with PME electrostatics. he only differences are that the oxygens have a charge of.8476 instead of.82 and the hydrogens have a charge of.4238 instead of.41. his small change in charge results in a viscosity of (kgm 1 s 1 ), which is closer to the experimental value of (kgm 1 s 1 ). 2 Smith and van Gunsteren report values of (kgm 1 s 1 ) and with the reaction-field, using the pressure fluctuation method. 18 he difference is caused by the error in the pressure fluctuations due to the cut-off treatment of the electrostatics, as was discussed above. he ratios of the values for the pressure fluctuation and the periodic perturbation method are the same as for SPC water. VIII. COCLUSIOS We have compared four methods known from literature for calculating the shear viscosity from an MD-simulation. he methods differ in applicability, ease of use and computational cost. For a Lennard-Jones fluid, which has only short-range interactions, all four methods give reliable results. With a more complex liquid, such as water, both equilibrium methods have some problems. he pressure fluctuations method requires accurate atomic forces, since these forces determine the virial and thus contribute to the pressure. o get accurate forces, the simulation should be performed in double precision. When Coulomb interactions are present, the long-range interactions should be treated accurately. Using a cut-off method, with a cut-off distance of.9 or 1.4 nm results in huge artifacts in the pressure fluctuations. his causes the calculated viscosity to be too high. A reaction-field reduces the artifacts, but only a long-range electrostatics method, such as PME, solves the problem completely. A second problem is choosing the period over which to average the integral of the pressure. his integral is very inaccurate for longer times. In general the averaging period can only be chosen after visual inspection. he transverse current autocorrelation method is a more direct approach, since it is based on the decay of correlation in the motion of particles, which is directly related to the viscosity. his makes the method insensitive to the type of electrostatics treatment. A fit to an analytical transverse current autocorrelation function is required to obtain the viscosity. he Lennard-Jones fluid follows the behavior of the avier Stokes equation. A phenomenological correction using a simple memory function improves the fit slightly. he behavior of water is more complicated, even with the memory function the fit is not optimal. A more complex analytical model is needed to describe the behavior of all liquids. hus far only the binary collision contribution to the transverse current correlation function has been derived. 21 Another problem is that the obtained viscosity depends on the wavelength, so extrapolation to infinite wavelength is required. he periodic perturbation method is similar to the transverse-current autocorrelation method. he difference is that, instead of using internal fluctuations, a shear is induced. his allows for a larger signal to noise ratio, while the shear

10 216 J. Chem. Phys., Vol. 116, o. 1, 1 January 22 Berk Hess rate can be chosen small enough that the structure and dynamics of the liquid are not disturbed. For a given simulation setup all relevant quantities can be estimated beforehand, including the simulation time required to reach a certain accuracy. he simulation box can be chosen to have an elongated shape such that the wavelength dependence of the viscosity becomes negligible. hus only one simulation is required. Also a larger box produces better statistics. o obtain an accurate viscosity the shear rate should be an order of magnitude smaller than the inverse of the correlation time in the liquid. Under such conditions the method is still computationally more efficient then both equilibrium methods. he SLLOD method has the same accuracy as the perturbation method at an equal average shear rate and equal computational time. However, it has several disadvantages. he equations of motion are non-hamiltonian and the system has an overall rotation. But more importantly, the velocity is prescribed, which generates forces, instead of forces generating a velocity profile. In simple liquids this is not an issue, but for more complex ones it might become a problem. here is also the practical inconvenience of sliding boundary conditions. hus the periodic perturbation method is the method of choice, as it is insensitive to the electrostatics treatment and just as efficient as the SLLOD method, while it does not share the disadvantages of the SLLOD method. APPEDIX: ERROR ESIMAIO In this appendix we derive the standard error in the mean of a correlated fluctuating quantity. Consider an observable x, which fluctuates in time around an average value c: S i1 i/ xtdt 1 xtdt i1/ 2. A4 When the block length is shorter than the correlation time of f, the error estimate will be too low. When is much longer than the correlation time, the error estimate will be almost exact. he expectation of S 2 () can be calculated analytically: S i1 1 i/ i1/ 1 i1 xtdt 1 xtdt 2 i/ 2 f tdt i1/ A5 1 f tdt 2 A6 1 1 / 2 f tdt 1 f tdt 2 A / / f tfudu dt xtc f t, A1 1 2 f tfu du dt. A8 f t. A2 We want to estimate the error in the estimate of c, which we obtain by averaging x(t) from t to. he optimal error estimate can be made when the autocorrelation of f is known: 1 2 xtdtc 2 1 f tdt f tfu du dt. A3 When it is unknown, the autocorrelation can be fitted to a given functional form. However, usually autocorrelation functions have long tails, which complicates the fitting. It is better to fit an integral property of f. One integral property is the error estimate from a block average. he data can be divided into blocks, over which the average is calculated. When the block averages are considered to be independent, a standard error estimate S can be calculated by dividing the standard deviation of these averages by : So the expectation can be calculated when the autocorrelation of f is known. When the autocorrelation is exponential, f tfu exp 2 tu, A9 the expectation of S 2 () appears to be S exp 1 A1 1 exp 1 1 exp, 1 A exp 1 O. A12 When is larger than the correlation time, the last term can be neglected. When is of the order of the observation time is not sufficient to estimate the error. We can also write the error estimate in terms of the block length t/:

11 J. Chem. Phys., Vol. 116, o. 1, 1 January 22 Shear viscosity from molecular dynamics simulations FIG. 8. Error estimates for 1/ for the 6912 SPC system with cut-off electrostatics and A.5 nm/ps 2. he analysis was done from 5 to 2 ps 1951 data points. he standard deviation of 1/ is 253. All curves are plotted as a function of the block length, which is 195/ ps, where is the number of blocks. he solid curve is the error estimate S() assuming the blocks are independent expression A4. he dashed curve is the fit using E(t) expression A16. he coefficient of the slow exponential is negative; this is due to bad statistics when using a small number of blocks. o obtain a more accurate error estimate, we fitted again with a single exponential, this is the dot dashed curve (13.3 ps). his gives a standard error estimate for 1/ of S 2 t 22 1 t exp 1 O t. A13 A single exponential autocorrelation can usually be fitted easily, but problems arise when there is a second, longer correlation time present in f : 1 f tfu 2 exp tu 1 exp tu. A14 2 Usually one decay time, which we choose to be 2, is at least one order of magnitude larger than the other and is very close to 1. he expectation now becomes S 2 t t exp t t exp t O O 2 2. A15 When we neglect the last 2 terms, we obtain a relatively simple function E which depends on as 1/: E 2 t t exp t t exp t 2 1. A16 he optimal error estimate Eq. A3 is given by the limiting value of E(t): lim t 217 Et A17 o estimate the error one needs to estimate,, 1 and 2. he standard deviation of x(t) over time t to provides the optimal estimate for. he other 3 parameters can be obtained from a fit of S(t) to E(t). here are restrictions on the range of the fitting parameters: should be between and 1 and 1 and 2 should be larger than. When the longest correlation time, 2, is longer than the averaging interval and (1) 2 is not negligible compared to 1, there is not enough statistics to estimate the error. An example is given in Fig B. J. Palmer, Phys. Rev. E 49, G. Ciccotti, G. Jacucci, and I. R. McDonald, J. Stat. Phys. 21, he name SLLOD originates from the use of the transposed Doll s tensor, the dyadic product qpof positions and momenta, which was named after the Kewpie Doll by Hoover. 4 D. J. Evans and G. P. Morriss, Statistical Mechanics of onequilibrium Liquids Academic, London, M. P. Allen and D. J. ildesley, Computer Simulations of Liquids Oxford Science Publications, Oxford, G. Arya, E. J. Maginn, and H.-C. Chang, J. Chem. Phys. 113, R. Zwanzig, Annu. Rev. Phys. Chem. 16, D. Forster, Hydrodynamics, Fluctuations, Broken Symmetry and Correlation Functions Benjamin/Cummings, Reading, MA, J. P. Boon and S. Yip, Molecular Hydrodynamics Dover, ew York, H. J. C. Berendsen, J. P. M. Postma, W. F. van Gunsteren, A. Diola, and J. R. Haak, J. Chem. Phys. 81, H. J. C. Berendsen, ransport properties computed by linear response through weak coupling to a bath, in Computer Simulations in Material Science, edited by M. Meyer and V. Pontikis Kluwer, Dordrecht, 1991, pp E. Lindahl, B. Hess, and D. van der Spoel, J. Mol. Model. Electronic Publication 7, H. J. C. Berendsen, J. P. M. Postma, W. F. van Gunsteren, and J. Hermans, Interaction models for water in relation to protein hydration, in Intermolecular Forces, edited by B. Pullman Reidel, Dordrecht, 1981, pp S. Miyamoto and P. A. Kollman, J. Comput. Chem. 13, R. O. Watts, Mol. Phys. 28, M. eumann and O. Steinhauser, Mol. Phys. 39, Darden, D. York, and L. Pedersen, J. Chem. Phys. 98, P. E. Smith and W. F. van Gunsteren, Chem. Phys. Lett. 215, H. J. C. Berendsen, J. R. Grigera, and. P. Straatsma, J. Phys. Chem. 91, R. C. Weast, in CRC Handbook of Chemistry and Physics CRC Press, Boca Raton, R. K. Sharma, K. ankeshwar, K.. Pathak, S. Ranganathan, and R. E. Johnson, J. Chem. Phys. 18,